Midterm I—Form B
Fall 2004
Economics 203
Instructor: Petry
Name_____________
SSN______________
Before beginning the exam, please verify that you have 13 pages with 40 questions in
your exam booklet. You should also have a decision-tree and formula sheet provided
by your TA. Please include your full name and Net-ID on your bubble sheets. Good
luck!
1. Consider a positively skewed mound-shaped distribution. Which of the following is
true?
a. mean < median < mode
b. median < mode < mean
c. mode < mean < median
d. mode < median < mean
e. median < mean < mode
2. Which of the following distributions is NOT symmetric about its mean?
a. normal distribution
b. standard normal distribution
c. t-distribution
d. chi-squared distribution
e. more than one of the above distributions is not symmetric about its mean.
3. Suppose that you want to estimate what share of the students got more than 70 points
on their Econ 203 midterm exam. Moreover, you want your estimate to be within
0.1 of its true population value with 90% confidence. The smallest sample size
required to accomplish this is:
z.1 = 1.28
a.
b.
c.
d.
e.
z.05 =1.645
z.025 = 1.96
40
41
59
67
68
4. Which of the following statements is NOT true concerning confidence intervals?
a. there are always two tails outside the confidence interval
b. they always contain the relevant population parameter
c. they always contain the relevant point estimate
d. they can sometimes be used for hypothesis testing
e. all of the above statements are true of confidence intervals
1
5. One of your fellow students suspects that Canadians spend more time reading
newspapers than Americans. To test her claim, she gathers a sample of the average
number of hours per week spent reading newspapers for 100 students at the
University of Saskatoon and does the same for 100 students at the University of
Illinois. Assuming that her choice of tests is limited to the following, which test
should she use to examine her claim?
a.
b.
c.
d.
e.
Z-test for difference in proportions
Z-test for difference in means
paired sample t-test for difference in means
t-test for difference in means assuming unequal variances
F-test for difference in variances
Use the following information to answer the next two questions (#6-7)
You just moved to London and since you are a big soccer fan, you would like to choose a
local team that you will root for. You live close to the Chelsea FC stadium, but the
problem is that you would like to be a fan of a team that wins most of the time. So,
although Chelsea’s stadium is closer than Arsenal’s, you are willing to sacrifice and
become an Arsenal fan if Arsenal wins more than 10% more than Chelsea. Last year,
Arsenal (sample 1) won 42 out of 56 games, while Chelsea (sample 2) won 30 out of 50
games.
6. What is the relevant point estimate?
a. 0.05
b. 0.15
c. 0.554
d. 0.679
e. 1.662
7. Ignoring the results of the previous question, suppose that the test statistic is -1.735.
What is the p-value for the above test?
z0.1=1.28
a.
b.
c.
d.
e.
z0.05=1.645
z0.025=1.96
less than 0.025
less than 0.05 but more than 0.025
less than 0.5 but more than 0.05
less than 0.95 but more than 0.5
less than 0.975 but more than 0.95
8. Which of the following is always true?
a. the covariance of a variable with itself is always equal to one
b. the coefficient of correlation of a variable with itself is always equal to one
c. the covariance between two variables is always positive
d. the correlation coefficient between two variables is always positive
e. the covariance between two variables can never be zero
2
9. The math placement exam scores for freshmen at the University of Illinois are
normally distributed with mean 75 and standard deviation 10. What is the probability
of seeing a score greater than 90 or a score less than 60? (Note: you should rely on the
probabilities given below to answer this question.)
(0 < z < 1.5) =0 .433
a.
b.
c.
d.
e.
P (-2.5< z < 0) =0 .494
0.067
0.062
0.134
0.124
0.866
Use the following information to answer the next two questions (#10-11)
Your high-school friend claims that the Northwestern football team has been fairly
consistent in the last 20 years in terms of the numbers of wins each season. Illinois, he
adds, has been more unpredictable. There are years when Illinois would win almost all
the games, but there were seasons where it would win only 2 games per season. So, you
decide to gather data and test your friends claim that the Northwestern football team
(sample 1) has been more consistent than the Illinois football team (sample 2) in the past
20 years. The table below contains the summary statistics of your data.
Mean
Standard deviation
Number of observation
Northwestern
5
14
20
Illinois
7
20
20
10. Given that the relevant critical value is 0.421, what would be your test statistic and
the conclusion for this test?
a. -2.317 and you would reject the null hypothesis
b. -0.366 and you would fail to reject the null hypothesis
c. 0.366 and you would reject the null hypothesis
d. 0.490 and you would fail to reject the null hypothesis
e. 0.490 and you would reject the null hypothesis
11. If, on the other hand, you wanted to test whether Illinois has been a better team (more
wins) than Northwestern in each of the past 20 years, which test would you perform?
a. t-test for difference in means assuming equal variances
b. Z-test for difference in means
c. Z-test for difference in proportions
d. F-test for difference in variances
e. paired sample t-test for difference in means
3
12. Suppose you’re considering whether to pursue a PhD in economics at the University
of Illinois, and you want to know whether being a graduate student in economics at
Illinois is really as much fun as you’ve heard it is. You interview 120 people who
were graduate students in economics at Illinois, and find that 80 of them thought it
was really fun. A 95% confidence interval for the share of graduate students who
thought it was really fun would be:
z0.1=1.28
a.
b.
c.
d.
e.
z0.05=1.645
z0.025=1.96
[0.524, 0.809]
[0.582, 0.751]
[0.507, 0.826]
[0.466, 0.867]
[0.422, 0.911]
13. Which of the following statements is true?
a. in a test on the difference of means between two populations, which
population we choose as population one will not affect the relevant point
estimate
b. in a test on the difference of means between two populations, which
population we choose as population one will not affect the test statistic
c. in a test on the difference of means between two populations, which
population we choose as population one will not affect the critical value
d. in a test on the difference of means between two populations, which
population we choose as population one will not affect the conclusion of the
test
e. all of the forgoing statements are false
14. Chicago Cubs, a baseball team, is considering trading Sammy Sosa because he has
not been hitting as many home runs as the management would like. The club will buy
a new player only if they find one that can hit at least 10 more home runs than
Sammy Sosa hits in a season. They have their eyes on Gary Sheffield of the New
York Yankees. The Cubs management gathers data on 18 seasons that Sammy Sosa
has played, and during that period he hit an average of 17 home runs per season with
the standard deviation of 12, while is his 19 years of play, Gary Sheffield hit an
average of 27 home runs per season with the standard deviation of 12. What decision
should they make?
a. reject the null hypothesis and trade Sammy Sosa for Gary Sheffield
b. reject the null hypothesis and do not trade Sammy Sosa for Gary Sheffield
c. fail to reject the null hypothesis and trade Sammy Sosa for Gary Sheffield
d. fail to reject the null hypothesis and do not trade Sammy Sosa for Gary
Sheffield
e. there is not enough information to say anything informative about the claim
4
Use the following information to answer the next two questions (#15-16)
A test preparation service claims that at least 10% of the students who take their course
get a perfect score on the math section of the GRE. To investigate this claim, you
sampled 200 people who used this service and found that 28 got a perfect score on the
math section of the GRE.
15. What is the appropriate test statistic for testing the company’s claim?
z0.1=1.28
a.
b.
c.
d.
e.
z0.05=1.645
z0.025=1.96
1.417
1.558
1.630
1.886
2.074
16. Ignoring the result of the previous question, assume that the correct test statistic was
1.659. Suppose that you want the probability of Type I error to be no more than .05.
What should you conclude in this case?
a. fail to reject the null hypothesis and conclude that the test preparation
company can claim that at least 10% of their students get a perfect score on
the math section
b. fail to reject the null hypothesis and conclude that the test preparation
company cannot claim that at least 10% of their students get a perfect score on
the math section
c. reject the null hypothesis and conclude that the test preparation company can
claim that at least 10% of their students get a perfect score on the math section
d. reject the null hypothesis and conclude that the test preparation company
cannot claim that at least 10% of their students get a perfect score on the math
section
e. there is not enough information to say anything informative about the claim
17. An education researcher is interested in the distribution of tuition bills for private
schools in Chicago. She collects the following data on private schools in Chicago and
reported the following values in thousands of dollars.
School
$(1000)
1
7
2
4
3
11
4
5
5
11
6
5
What are the mean, median, and mode, respectively?
a. 7.3, 6, 5
b. 7.2, 5, 5
c. 7.3, 5.5, 5
d. 7.2, 5.5, 11
e. none of the above
5
7
6
8
5
9
15
10
3
Use the following information to answer the next two questions (#18-19)
One of the teaching assistants claims that the grades on the first exam last semester were
closer together than they had been in the past. To test if this claim is true, he looks at a
sample of 12 exams from last semester:
44, 45, 44, 46, 47, 46, 47, 49, 44, 42, 48, 43
Historically, the variance of exam scores was 4.232.
20.975,11 = 3.82
20.025,11 = 21.92
20.975,12 = 4.40
20.025,12 = 23.34
18. What is the value of the appropriate test statistic for examining the TA’s claim?
a. 9.415
b. 9.910
c. 10.432
d. 10.981
e. 11.559
19. What is the lower level of a 95% confidence interval for the variation in grades last
semester?
a. 2.096
b. 2.232
c. 4.076
d. 12.805
e. 11.117
20. Which of the following is implied by the Central Limit Theorem?
a. if a random sample is drawn from any population then the sampling
distribution of the sample mean is approximately normal for any sample size
b. if a random sample is drawn from any population then the sampling
distribution of the sample mean is approximately normal for a sufficiently
large sample size
c. if a random sample is drawn from any population then the sampling
distribution of the sample mean is approximately standard normal for any
sample size
d. if a random sample is drawn from any population then the sampling
distribution of the sample mean is approximately standard normal for a
sufficiently large sample size
e. if a random sample is drawn from a normal population then the sampling
distribution of the sample mean is approximately normal for a sufficiently
large sample size
6
Use the following information to answer the next two questions (#21-22)
You are an analyst for a shipping company and you are asked to determine whether
workers who have been on the job more than 20 years have different productivity than
workers in general. Data is collected on how many packages a day are shipped from a
random sample of 50 workers who have been on the job more than 20 years. The mean
and standard deviation calculated from this sample are 267 and 45, respectively. From
other research, you know that the average for workers in general is 280.
21. What can you say about the test statistic?
a. it equals 2.043 and follows a t distribution with 49 degrees of freedom
b. it equals 2.043 and follows a t distribution with 50 degrees of freedom
c. it equals -2.043 and follows a z distribution with 49 degrees of freedom
d. it equals -2.043 and follows a t distribution with 49 degrees of freedom
e. it equals -2.043 and follows a t distribution with 50 degrees of freedom
22. If you were testing this hypothesis at the 1% significance level, what would you
conclude?
z0.05=1.645
t0.05,49= 1.68
t0.01,49 = 2.40
z0.025= 1.96
t0.025,49 =2.01
t0.005,49=2.68
a. fail to reject the null hypothesis and conclude there is not enough evidence to
claim that there is a difference in the productivity of older workers
b. reject the null hypothesis and conclude there is not enough evidence to claim
that there is a difference in the productivity of older workers
c. fail to reject the null hypothesis and conclude that older workers are less
productive
d. reject the null hypothesis and conclude that older workers are less productive
e. reject the null hypothesis and conclude that productivity is different for older
workers
23. Suppose you are a Congressional candidate trying to allocate resources between
advertising in the Daily Illini or in the student newspaper at Illinois State. You think
that the Daily Illini reaches fewer students. To test this claim, which could be a
proper null and alternative hypotheses (data is the form of the average number of
students who read the Daily Illini followed by number of students who read the ISU
paper on 25 randomly selected days during the academic year)?
a. H0: μ1 ≥ μ2; H1: μ1 ≠ μ2
b. H0: μ1 = μ2; H1: μ1 ≤ μ2
c. H0: μ1 = μ2; H1: μ1 < μ2
d. both a and c
e. a, b, and c
7
Use the following information to answer the next four questions (#24-27)
You are enrolled in a gender studies class at the UIUC and you professor has given you
an assignment to test a claim by a renowned researcher who claims that female students
spend more money on textbooks than male students. You do a survey on campus and
gather data on 10 female (sample 1) and 12 male (sample 2) students. Female students
spend, on average, $450 per semester on books, with a standard deviation of $75. Male
students, on the other hand, spend an average of $405 on books per semester, with a
standard deviation of $45. Moreover, from previous studies you know that variances of
the two populations are the same.
24. To test the researcher’s claim, you would perform a certain hypothesis test. What
would your null and alternative hypotheses look like?
a. H0: μ1- μ2 ≥ 0; H1: μ1- μ2 < 0
b. H0: μ1- μ2 ≤ 0; H1: μ1- μ2 > 0
c. H0:  12  22 ≤ 1; H1:  12  22 > 1
d. H0:  12  22 ≥ 1; H1:  12  22 < 1
e. H0:  12   22 ≤ 0; H1:  12   22 > 0
25. What would be the value of the point estimate?
a. 1.667
b. 1.741
c. 13.416
d. 30
e. 45
26. Ignoring the answer that you got in the previous question, suppose that the correct
test statistic was 2.101. What is your conclusion about the researcher’s claim at the
5% level of significance?
t0.05,14=1.761
t0.025,14=2.148
t0.01,14=2.624
t0.05,20=1.725
t0.025,20=2.086
t0.01,20=2.528
a. reject the null hypothesis and conclude that female students spend less on
textbooks than male students
b. fail to reject the null hypothesis and conclude that female students spend more
on textbooks than male students
c. reject the null hypothesis and conclude that male students spend more on
textbooks than female students
d. reject the null hypothesis and conclude that female students spend more on
textbooks than male students
e. fail to reject the null hypothesis and conclude that there is insufficient
evidence to conclude that female students spend more on textbooks than male
students
8
27. Now suppose that you are skeptical about the equal population variances assumption
used throughout this set of questions. So, you perform an appropriate test. What is
your conclusion at the 10% level of significance?
F0.025, 9, 11=3.588
F0.025, 10, 12=3.374
F0.05, 9, 11=2.896
F0.05,10, 12=2.754
F0.1, 9, 11=2.274
F0.1, 10, 12=2.188
a. there is insufficient evidence to conclude that the population variances are
equal, so the above assumption of equal variances is correct
b. there is insufficient evidence to conclude that the population variances are
equal, so I would have to re-evaluate the above answers
c. there is enough evidence to conclude that the population variances are unequal,
so the above assumption of equal variances is correct
d. there is enough evidence to conclude that the population variances are unequal,
so I would have to re-evaluate the above answers
e. there is insufficient evidence to conclude that the population variances are
unequal, so the above assumption of equal variances is correct
28. You want to test whether the population mean for widgets is greater than the
population mean for snidgets, assuming unequal variances. So, taking widgets as
population one, you crank up Excel and get the following output:
t-Test: Two-Sample Assuming Unequal Variances
Mean
Variance
Observations
Hypothesized Mean Difference
df
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail
Widgets
96.66
314.512
10
0
18
-0.578
0.285
1.330
0.570
1.734
Snidgets
101.39
354.75
10
What is the p-value associated with the test you intended to perform?
a. 0.285
b. 0.430
c. 0.570
d. 0.715
e. there is not enough information provided to determine the p-value
9
29. The weights of the Illini football team were collected and it was found that the mean
weight was 230 pounds with a standard deviation of 25 pounds. Approximately, what
fraction of the team weighs more than 305 pounds?
a. 99%
b. 5%
c. 0.005%
d. 0.5%
e. -98.5%
30. Suppose that we are comparing two populations. We have a sample of size 10 from
each, and we want to test whether the mean of the first population is greater than the
mean of the second. Suppose that we have no reason to believe that the population
variances are the same, and the sample variances are 3 and 4. The test statistic that we
would use would follow?
a. a Z-distribution with 9 degrees of freedom
b. a t-distribution with 9 degrees of freedom
c. a t-distribution with 18 degrees of freedom
d. a t-distribution with 16 degrees of freedom
e. a t-distribution with 20 degrees of freedom
Use the following information to answer the next two questions (#31-32)
Suppose that you want to investigate a relationship between the years of schooling and
salary. You gather data on 6 people and the results are presented in the following table:
Years
Salary($1000)
12
16
14
17.5
12
22
16
35
18
62
20
42
31. Calculate the sample covariance and correlation coefficient between years of
schooling and salary.
a.
b.
c.
d.
e.
cov(x,y) = 39.1, r = .810
cov(x,y) = .810 , r = 46.933
cov(x,y) = 46.9 , r = .810
cov(x,y) = 46.9 , r = .889
none of the above
32. Using the answers from the previous question, which of the following conclusions
can you make about the relationship between years of schooling and salary?
a. the covariance value tells us that there is a strong positive linear relationship
between years of schooling and salary
b. the correlation coefficient tells us that the slope of the line that defines the
relationship between the two variables is positive
c. the correlation coefficient is the value of the slope of the line that defines the
relationship between the schooling and salary
d. both a and b
e. all of the above
10
33. It is believed that the variance of a particular process is 0.1. You have a sample from
this process where the variance is 0.15. Based on this sample, you would like to claim
that the prevailing belief is in error. You propose to test the alternative hypothesis
σ2>0.1 against the null hypothesis σ 2 = 0.1. Your study partner thinks that you should
test the alternative hypothesis σ 2 ≠ 0.1 against the null hypothesis σ 2 = 0.1. Which of
the following statements is true, assuming a 10% probability of a Type 1 error?
a. it will require more extreme values of the test statistic to reject the null
hypothesis if you use H1: σ2 > 0.1 than if you use H1: σ2 ≠ 0.1
b. it will require more extreme values of the test statistic to reject the null
hypothesis if you use H1: σ2 ≠ 0.1 than if you use H1: σ2 > 0.1
c. the choice of H1: σ2 > 0.1 versus H1: σ2 ≠ 0.1 will not affect how extreme a
test statistic you will need to reject the null hypothesis
d. you do not have enough information to make any statement about the effect of
choosing H1: σ2 > 0.1 versus H1: σ2 ≠ 0.1 on how extreme a test statistic you
will need to reject the null hypothesis
34. Prof. Petry claims that people who go to the review session before the exam do better
on the exam than people who don’t go to the review session. To test his claim,
Marcus collected data on 75 students who went to the review session (sample 1) and
58 students who did not go to the review session (sample 2). The mean score of the
students who went to the review session was 80 with the standard deviation of 12,
while the mean score of the students who did not go to the review session was 84
with the standard deviation of 12. From the tests listed below, which would be the
most appropriate for Marcus to perform?
a. t-test for difference in means assuming equal variances
b. Z-test for difference in means
c. Z-test for difference in proportions
d. 2-test for difference in variances
e. paired sample t-test for difference in means
35. All of the following statements are false, except
a. the variance of the sampling distribution of the sample mean is always equal
to the variance of the population
b. the sample standard deviation is the square root of the sum of squared
deviations from the population mean divided by the sample size minus one
c. the range is a measure of central tendency
d. the median is the value that falls in the middle of a set of ordered
measurements
e. both b and d
11
36. You want to find out whether the average income in Champaign is different than the
average income in Urbana. You gather data and the average income in Champaign
(sample 1) is $36,900 with standard deviation $14,200, while the average salary in
Urbana (sample 2) is $39,100 with standard deviation $16,800. You calculate the
95% confidence interval for the true difference in population means and it turns out to
be [-5543.453, 1143.451]. How do you proceed?
a. no data is provided on the numbers of observations for each sample, so I
cannot proceed
b. I cannot determine whether I should perform an equal or unequal variances ttest for difference in means since I cannot perform a F-test for the ratio of
variances
c. I would reject the null hypothesis and conclude that Champaign and Urbana
have the same average income
d. I would fail to reject the null hypothesis and conclude that there is insufficient
evidence to conclude that average incomes are different in Champaign and
Urbana
e. I would reject the null hypothesis and conclude that the average income in
Urbana is higher than in Champaign
37. In testing the difference between two population means assuming unequal variances,
your p-value was larger than .05, so you did not reject the hypothesis that the
difference in means was zero. You report that you are 95% confident that the means
are equal. Was this correct?
a. yes, this it what it means to be 95% confident
b. no, you do not have the basis to make this claim
c. it would be correct, if you knew that the population variances were equal.
d. it would be correct, if the population variances were known
e. regardless of what you know about the population variances, you do not have
enough information to make any statement about the relationship between the
population means
38. To test whether more than 50% of the UIUC students spend 40% or more of their free
time watching TV, your null hypothesis would be:
a. H0: p ≤ 0.40
b. H0: p > 50
c. H0: p = 0.40
d. H0: p ≤ 0.50
e. H0: p ≥ 0.50
12
39. Females on average spend more on clothing apparel than males. You want to find out
whether more females spend at least 1/3 of their income on clothing and apparel than
males do. Which of the following tests can you use?
a. pooled-variance t-test for difference in means
b. paired sample t-test for the mean difference
c. Z-test for difference in proportions
d. F-test for difference in variances
e. F-test for difference in means
40. Given the following p-values and their associated Z-scores, what is the last entry in
the table?
p-value
0.023
0.050
0.025
0.159
0.284
0.097
a.
b.
c.
d.
e.
Z-statistic
2
1.645
1.96
1
0.57
0.3
0.8
1.3
1.8
2.3
13
ANSWER KEY
1. D
2. D
3. E
4. B
5. D
6. B
7. E
8. B
9. C
10. D
11. A or E
12. B
13. D
14. D
15. D
16. C
17. E
18. E
19. B
20. B
21. D
22. A
23. C
24. B
25. E
26. D
27. E
28. D
29. C or D
30. C
31. C
32. B
33. D
34. A
35. D
36. D
37. B
38. D
39. C
40. C
14